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We study the asymptotic behavior of solutions to a monostable integro-differential Fisher-KPP equation, that is, where the standard Laplacian is replaced by a convolution term, when the dispersal kernel is fat-tailed. We focus on two different regimes. First, we study the long time/long range scaling limit by introducing a relevant rescaling in space and time and prove a sharp bound on the (superlinear) spreading rate in the Hamilton--Jacobi sense by means of sub- and supersolutions. Second, we investigate a long time/small mutation regime for which, after identifying a relevant rescaling for the size of mutations, we derive a Hamilton--Jacobi limit.